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1.
To undermine or belittle another person, especially on an intellectual level.
Fernando takes great pleasure in despouching those less intelligent than he.

Using sarcasm, it was easy for Juan to despouch Martinez in front of his friends.

"You would thank that, but you'd be wrong. You could not be more wrong. You could try, but you would not be successful" said Esteban, despouching Diego.
by Realph March 16, 2008
 
2.
To detract or negate the meaning of a given idea or clause.
His baseless argument was easily despouched using logic.
by Realph March 16, 2008